Legendre’s conjecture

A prime is a whole number greater than 1 whose only positive divisors are 1 and itself. Legendre's conjecture says there is always a prime strictly between two consecutive positive whole-number squares. Between $2^2=4$ and $3^2=9$, there are 5 and 7. Between 9 and 16, there are 11 and 13. Does at least one prime appear in every such interval, however large the squares become?[1]

Legendre's conjecture asks for a prime in every interval between consecutive positive integer squares:

$$\forall n\in\mathbb{Z}_{>0},\quad\exists p\text{ prime}:\quad n^2<p<(n+1)^2.$$

The inequalities are strict, and $n$ ranges over integers. The interval width $2n+1$ is of order the square root of its position near $n^2$. The claim concerns all of these intervals, rather than an average count or only those up to a computational threshold. Results between larger powers, or results assuming the Riemann hypothesis, must be assessed against these exact requirements.[1]

The answer is not proved for every pair.[1]

The February 2026 results concern intervals between larger powers and assume the Riemann hypothesis. Those wider, conditional statements do not prove the original square-interval conjecture.[1]

Primes have uneven gaps. This conjecture would put a limit on how long a stretch without a prime can last: it could never contain the entire interval between consecutive squares. Those intervals get wider, but increasing room alone does not prove a prime will be there.[1]

The conjecture concerns local prime distribution at a specified scale. An asymptotic description of how many primes occur up to a large boundary does not itself prevent one square interval from being empty. A resolution would establish or refute this concrete restriction on prime-free stretches; wider conditional intervals leave that restriction open.[1]